A214981 Number of terms in the greedy Lucas-and-Fibonacci representations of 1,2,...,n; partial sums of A214973.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 31, 33, 35, 37, 39, 41, 44, 46, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 66, 69, 71, 73, 76, 79, 81, 84, 85, 87, 89, 91, 93, 95, 98, 100, 101, 103, 105, 107, 109, 111, 114, 116, 118, 121, 124

Offset: 1

Author

Clark Kimberling, Oct 22 2012

Keywords

• nonn

Comments

For comparison with Zeckendorf (Fibonacci) representations, it is conjectured that the limit of A179180(n)/A214981(n) exists and is between 1.2 and 1.4.

Examples

			The basis is B = (1,2,3,4,5,7,8,11,13,18,21,29,34,47,55,...), composed of Fibonacci numbers and Lucas numbers. Representations of positive integers using the greedy algorithm on B:
   n  repres.   # terms  a(n)
   1   1        1        1
   2   2        1        2
   3   3        1        3
   4   4        1        4
   5   5        1        5
   6   5+1      2        7
   7   7        1        8
   8   8        1        9
   9   8+1      2       11
  10   8+2      2       13
  27   21+5+1   3       44
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.

Crossrefs

Cf. A000032, A000045, A179180, A214973, A214977, A214979, A214980, A214981.

Programs

Mathematica

(See the program at A214973.)

Extensions

Edited by Clark Kimberling, Jun 13 2020